All (vehicle) control systems have physical actuators, e.g., electrical motors, hydraulic servo valves, etc. These actuators all have rate limits due to limits in power supply, hydraulic pressure, etc. Thus, all control systems inherently include restrictions with regard to the rate at which a new command from the driver of the vehicle, i.e., a change in the input signal into the control system, can give rise to corresponding changes in the physical output signal from the control system. If the time derivative for the input signal exceeds a certain value, the time derivative for the output signal is limited in relation to the time derivative for the input signal. This limitation is known as the rate limitation of the control system. When very rapid changes in the input signal are executed, for example when the input signal to the control system consists of an excessively rapid (or large) sinusoidal signal, a phase shift occurs between the input signal and the output signal. That is, the output signal is subject to a time delay in relation to the input signal. This phase shift leads to impairment of the performance of the vehicle and, in the worst case, can give rise to instability.
In aircraft applications, a so-called PIO (Pilot Induced Oscillation) can occur when an excessively rapid change in the input signal causes the rate limitation of the control system to be exceeded. This can occur if an unforeseen circumstance causes the pilot to execute rapid and large movements with the control stick of the aircraft. The phase shift which occurs because of the rate limitation of the control system amplifies the oscillations. In the worst case these oscillations become divergent, which can result in loss of control over the movements of the aircraft.
The aforementioned rate limitation is more noticeable in aircraft which constitute a so-called unstable system. In this type of aircraft, the rudders of the aircraft are affected not only by the signals from the pilot, but also by stabilization signals generated in the control system, which are dependent on values obtained from sensors at different points in the aircraft.
One way of reducing the aforementioned problems in control systems with rate limitation involves the introduction of phase compensation when the rate limitation is active. Such phase compensation must meet the following requirements:
A. reduce the phase retardation in the case of sinusoidal input signals;
B. minimize the dynamic retardation for rapid ramp and steps;
C. provide the same input and output signal when the input signals are sufficiently slow.
Two different methods of executing phase compensation in control systems with rate limitation are previously disclosed in Buchholz, J. J. (1993): "Time delay induced by Control Surface Rate Saturation", Zeitschrift f. Flugwissenschaften und Weltraumforschung, Springer Verlag, Vol. 17, pp. 287-293; A'harrah, R. C. (1992); "Communique with DLR and others", NASA HQ, Washington DC, July 14th, 1992; and Chalk, C. R. (1992): "Study of a Software Rate Limit Concept", Calspan Flight Research Memorandum 635, Buffalo, N.Y. These methods use logical conditions (if-then-else) to establish whether a phase compensation requires to be executed in the control system. However, these conditions call for a jump to be made between at least two different dynamic behaviors for the respective methods. Thus, input signals can always occur which give rise to an undesired output signal. For this reason, none of these methods is suitable to be implemented in a control system with rate limitation.